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I'm just reading about the group SL(2,C). In the book that I'm using(Jones, groups reps and physics), he defines a 2x2 matrix from a generic 4 vector [tex]v_{\mu}[/tex] and a vector [tex] \sigma_{\mu}:=(1,\vec{\sigma})[/tex], as [tex] V:=v_{\mu}\sigma^{\mu}[/tex]

He nows wants to invert this equation to solve for [tex]v_{\mu}[/tex], and he suggests tracing with another vector of matrices defined as [tex] \tilde{\sigma_{\mu}}:=(1,-\vec{\sigma})[/tex], and he obtains [tex] v_{\mu}=\tfrac{1}{2}Tr(\tilde{\sigma_{\mu}}V)[/tex]

I can't seem to get this, starting with [tex] V:=v_{\mu}\sigma^{\mu}[/tex] and then multiplying by [tex] \tilde{\sigma_{\nu}} [/tex], leads to [tex] \tilde{\sigma_{\nu}}V:=v_{\mu}\sigma^{\mu}\tilde{\sigma_{\nu}}[/tex]

Now I'm not sure what indices I'm supposed to trace with?

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# Group SL(2,C)

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